If you're referring to the programs actually still being on your hard drive, they will certainly stay. This has been the case for every OS.
If, however, you're talking about compatibility, it does vary from version to version. The majority of programs aren't broken by a new OS, but there are always one or two that are. This is as true today as it was 20 years ago when we were preparing for System 7. Looking at older Macs, you'll find some System 6-era programs that didn't run properly on System 7. Many of these were irrelevant/obsolete by 1991 or were from smaller developers.
The one exception is when you're going from OS 9 to OS X and aren't installing Classic. Your pre-OS X programs will only run with Classic installed, and if you've got a newer Mac or a newer version of OS X, Classic won't run.
Another exception goes for software specifically designed for a certain system version. UnScrolly and ChooseCDEV, two useful System 6 control panels, served no purpose under System 7 because the control panel changed.
It's also worth pointing out that new operating system versions sometimes will break viruses.
Downgrading works in a similar way to upgrading. If you're going from System 7 to System 6 (pretty easy to do after you've tried System 7 on a 2MB Mac Classic for a few days), you'll want to make sure you have software that runs on System 6. Some titles require System 7. Manuals and boxes will typically state what is required, as will the read me files that come with public domain software.